Need Help on a data analysis problem

I would like to preface this by saying that I am new to Excel and that I have tried looking up a solution to my problem online. Whilst I will continue trying to find a solution on my own I figured there is no harm posting here.

I will first describe the problem. My data consists of 20 columns. On the first row we have names, on the next thousand rows we have numbers. Each one of these thousands of rows represents an event, wherein the name with the highest number gets first place, the name with the second highest number gets second place and so on. My goal is to find how many times did name 1 get first place, repeat for name 2 and so on and then repeat that for every other place.

I thought if I could analyze one row to give me first the name with the highest number on that row and then the name with the second highest number and so on, I would have 20 columns wherein the first column would have the name with the highest number, the second column would have the name with the second highest number and so on. I would then expand that analysis down 999 rows and then I would have in that first column the names of all the first place finishers. I could then read that column and get the results, say name 1 had x number of first places and so on, then repeat for every other column. Unfortunately my limited Excel knowledge didn't allow me to do that.

I would appreciate any help either with making my solution work or with coming up with a better way to solve this. One less pressing issue is to find a way of dealing with ties but I have some ideas on that.

So, basically, you want to sort each row from large to small, is what I am gathering from your description. Here's how I would expect to proceed (I hope you don't mind if I use column numbers rather than letters, since I don't want to try to convert 20+ columns to letters).

1) The LARGE() function can easily be used to get the numbers in order. I would enter 1, 2, 3,...,20 in columns 21 to 40 of row 1. Then enter =LARGE($A2:$T2,U$1) in U2, note the mix of relative and absolute references, and copy down and across as far as needed. The largest number in each row is now in column 21 (U), the second largest in column 22 (V), and so on.
2) A MATCH() function (exact match option) will readily determine which column contained each number. =MATCH(U2,$A2:$T2,0) (I think this would be in column AO -- column 41). Again, note the mix of relative and absolute references, and copy/paste/fill down and across as far as needed.

2a) At this point, you can analyze the column numbers. If you decide you must analyze on name, then you can nest the MATCH() function in 2 inside of an INDEX() function to get names. =INDEX($A$1:$T$1,MATCH(...)). I'm inclined to think that the numeric analysis might be more efficient, then extract names with the INDEX() function at the end of the data analysis, but it would really be up to you.

That should be a good start, any questions or difficulty with implementing those formulas?